The Combinatorics of Motzkin Polyominoes

Abstract

A word $w=w_1\cdots w_n$ over the set of positive integers is a Motzkin word whenever $w_1=\texttt{1}$, $1\leq w_k\leq w_{k-1}+1$, and $w_{k-1}\neq w_{k}$ for $k=2, \dots, n$. It can be associated to a $n$-column Motzkin polyomino whose $i$-th column contains $w_i$ cells, and all columns are bottom-justified. We reveal bijective connections between Motzkin paths, restricted Catalan words, primitive Łukasiewicz paths, and Motzkin polyominoes. Using the aforementioned bijections together with classical one-to-one correspondence with Dyck paths avoiding $UDU$s, we provide generating functions with respect to the length, area, semiperimeter, value of the last symbol, and number of interior points of Motzkin polyominoes. We give asymptotics and closed-form expressions for the total area, total semiperimeter, sum of the last symbol values, and total number of interior points over all Motzkin polyominoes of a given length. We also present and prove an engaging trinomial relation concerning the number of cells lying at different levels and first terms of the expanded $(1+x+x^2)^n$.

Figures (11)

• Figure 1: Motzkin words of length 5 and their associated Motzkin polyominoes.
• Figure 2: Bijection between Motzkin paths and Motzkin words.
• Figure 3: Bijection between Motzkin words and primitive Łukasiewicz paths.
• Figure 4: Motzkin polyominoes whose last column is of height $1$.
• Figure 5: Motzkin polyominoes whose last column is of height $\geq 2$.

Theorems & Definitions (30)

• Theorem 3.1
• proof
• Corollary 3.2
• Corollary 3.3
• Theorem 4.1
• Theorem 4.2
• proof
• Theorem 4.3
• Corollary 4.4
• Corollary 4.5